I've been thinking about open balls in $\mathbb{R}^n$ and whether the density of $\mathbb{Q}$ in $\mathbb{R}$ means that we can find open balls in $\mathbb{Q}^n$ to 'nest inside open balls in $\mathbb{R}^n$, if we take the 'centre' of the ball to be distinct. I've tried to formalise my question below, in an arbitrary dimension (the one-dimensional case seems more simple):

If we think of $\mathbb{R}^n$ endowed with the $\|\|_{\infty}$ norm, can we show that for any open ball around an arbitrary point $x \in \mathbb{R}^n$, $B(x,\epsilon)$ with $\epsilon>0$ given, that there is some $q \in \mathbb{Q}^n$, $\delta>0$ such that